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We study the distortion of one-sided and two-sided matching problems on the line. In the one-sided case, n agents need to be matched to n items, and each agent's cost in a matching is their distance from the item they were matched to. We propose an algorithm that is provided only with ordinal information regarding the agents' preferences (each agent's ranking of the items from most- to least-preferred) and returns a matching aiming to minimize the social cost with respect to the agents' true (cardinal) costs. We prove that our algorithm simultaneously achieves the best-possible approximation of 3 (known as distortion) with respect to a variety of social cost measures which include the utilitarian and egalitarian social cost. In the two-sided case, where the agents need be matched to n other agents and both sides report their ordinal preferences over each other, we show that it is always possible to compute an optimal matching. In fact, we show that this optimal matching can be achieved using even less information, and we provide bounds regarding the sufficient number of queries. 

We study the problem of designing voting rules that take as input the ordinal preferences of n agents over a set of m alternatives and output a single alternative, aiming to optimize the overall happiness of the agents. The input to the voting rule is each agent’s ranking of the alternatives from most to least preferred, yet the agents have more refined (cardinal) preferences that capture the intensity with which they prefer one alternative over another. To quantify the extent to which voting rules can optimize over the cardinal preferences given access only to the ordinal ones, prior work has used the distortion measure, i.e., the worst-case approximation ratio between a voting rule’s performance and the best performance achievable given the cardinal preferences. The work on the distortion of voting rules has been largely divided into two “worlds”: utilitarian distortion and metric distortion. In the former, the cardinal preferences of the agents correspond to general utilities and the goal is to maximize a normalized social welfare. In the latter, the agents’ cardinal preferences correspond to costs given by distances in an underlying metric space and the goal is to minimize the (unnormalized) social cost. Several deterministic and randomized voting rules have been proposed and evaluated for each of these worlds separately, gradually improving the achievable distortion bounds, but none of the known voting rules perform well in both worlds simultaneously. In this work, we prove that one can in fact achieve the “best of both worlds” by designing new voting rules, both deterministic and randomized, that simultaneously achieve near-optimal distortion guarantees in both distortion worlds. We also prove that this positive result does not generalize to the case where the voting rule is provided with the rankings of only the top-t alternatives of each agent, for t < m, and study the extent to which such best-of-both-worlds guarantees can be achieved.